Hilbert scheme of points on non-reduced nodal curves

We construct a stratification of the punctual Hilbert scheme of points on a non-reduced and nodal plane curve, $x^u{y}^v=0$. Each stratum is indexed by a new combinatorial object we define: a weak diagonal partition. The approach is based on introducing filtrations on ideals, together with a valuation adapted to the non-reduced structure, which allows us to analyze generators and their degrees of freedom in a systematic way. In particular, each stratum is affine when $u=1,2$; and each stratum is isomorphic to an algebraic torus times an affine space, $({\mathbb{C}}^{\ast }{)}^{m_1}\times {\mathbb{C}}^{m_2}$, when $u=v,v-1,v-2$. We consequently compute the Poincaré polynomials of the punctual Hilbert scheme of points on curves $x^u{y}^v=0$ when $u=1,2,v-2,v-1,v$. As an application, we prove the colored Oblomkov-Rasmussen-Shende conjecture for the Hopf link for $u=1,v$ arbitrary, showing that the Poincaré polynomial is the row-colored link homology up to change of variables.